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In this problem, we're going
to look at how to infer a
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continuous random variable from
a discrete measurement.
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And the continuous random
variable that we're interested
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in in this problem is q, which
is given by this PDF.
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It's 6q times 1 minus
q for a q between
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0 and 1 and 0 otherwise.
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And here is a graph of
what it looks like.
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It just kind of has
this curve shape.
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And it's symmetric.
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And it's peak is at 1/2.
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And the way to interpret
q is q is the
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unknown bias of a coin.
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So the bias of a coin, the
probability of heads of that
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coin is somewhere
between 0 and 1.
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We're not sure exactly
what it is.
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And here is our, say, prior
belief about how this random
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bias is distributed.
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And we're going try to infer
what this bias is by flipping
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the coin and observing whether
or not we got heads or tails.
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And because of that, the
measurement, or the
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observation that we
get, is discreet.
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Either we get heads,
or we get tails.
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And we model that using a
discrete random variable x,
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which is, in this case, it
turns out it's just a
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Bernoulli random variable,
either 0 or 1.
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And the model that we have is
that, if we knew what this
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bias q was, if we knew that it
was a little q, then this
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coin, I mean, it behaves as if
it was a coin with that bias.
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So the probability of getting
heads, or the probability that
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x equals 1, is just equal
to q, which is the bias.
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And the probability that it's
equal to 0 is 1 minus q.
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So that's just our model
of how the coin works.
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We can also write this another
way, as just more like a
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conditional PMF.
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So the conditional PMF of x,
given q of little x, is q, if
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x is 1, 1 minus q, if x equals
0, and 0 otherwise.
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Just a more compact way
of writing this.
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All right, so what we want to
do in this problem is find
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this conditional PDF.
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What is the conditional
PDF of q given x?
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So we observe what x is,
either a 0 or 1.
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And we want to know now, given
that information, given that
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measurement, what is the
new distribution of q
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the bias of the coin?
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And to do that, well, we
apply Bayes' rule.
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And remember, Bayes' rule, it
consists of several terms.
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The first one is the numerator,
which is our prior
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initial belief, so which is
just the regular PDF of q,
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times the conditional
PMF of x given q.
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All right, so because we have a
continuous variable that we
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want to infer from a discreet
measurement, we use this
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variation of Bayes' rule, where
we have a PDF here and a
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conditional PMF here.
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And the denominator is--
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well, it's the PMF of x.
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And of course, we can take
this PMF of x, this
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denominator, and expand it,
using the law of total
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probability where the PMF of x,
you can think of it as you
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can get x with a combination of
lots of different possible
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values of the bias q.
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And so we just calculate
all those
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possibilities and integrate.
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And what we want to integrate
here is q, so we want to
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integrate-- remember to keep
in mind the limits of
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integration.
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And this is just referenced
by the limits of what
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the PDF of q is.
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OK.
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All right, so now we're asked to
find what this value is for
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x equals to 0 or 1 and
for all values of q.
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And the values of q we care
about are the ones
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between 0 and 1.
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So let's focus on the two
different possibilities for x.
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So the first one is, let's
look at the case where x
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equals 1 first.
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And then now let's just
plug-in what all these
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different terms should be.
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Well, the PDF of
q we're given.
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And of course, we're looking
here at q between 0 and 1, so
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within that range.
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The PDF of q is just
6q times 1 minus q.
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And the conditional PMF of x
where we know that from our
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model, because we're looking at
the case where x equals 1.
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That conditional
PMF is just q.
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And the denominator is really
the same as the numerator,
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except you integrate it.
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So it's the integral from
0 to 1 of 6q times 1
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minus q times q dq.
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OK.
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And now we can simplify this.
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So under the numerator,
we have integral--
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sorry, 6q squared
times 1 minus q.
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And then the bottom we have
the integral of 6q squared
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minus q cubed, d cubed
from 0 to 1.
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And now this is just
some calculus now.
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So we still have the numerator
6q squared times 1 minus q.
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The denominator, we
have 2q cubed--
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that would give us the
6q squared term--
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minus 6/4 q to the fourth.
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And we integrate that
from 0 to 1.
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OK.
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And what does that give us?
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We get 6q squared 1 minus
q still on the top.
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And the bottom, we get--
well the 0--
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the case where it's
0, it's just 0.
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The case where it's 1, it's
2 minus 3/2, so it's 1/2.
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So really, it just becomes
12 q squared 1 minus q.
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And of course, this only true
when q is between 0 and 1.
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All right, so the case
where it's equal to
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1, we have our answer.
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And it turns out that, if
you plot this, what
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does it look like?
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It looks something like this
where the peak is now at 2/3.
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So how do you interpret this?
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The interpretation is that what
you have is you observe
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that you've got, actually,
heads on this toss.
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So that is evidence that
the bias of the coin
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is relatively high.
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So it's relatively more likely
to get heads with this coin.
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So q, in that case, you would
believe that it's more likely
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to be higher than 1/2.
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And it turns out that, when you
go through this reasoning
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and the Bayes' rule, what
you get is that
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it looks like this.
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And the peak is now at 2/3.
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And you can repeat this exercise
now with the case
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where x is 0.
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So when x is 0, you still get
the same term here, 6q 1 minus
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q, but the conditional PMF
is now the-- you want the
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conditional PMF when x equals
0, which is now 1 minus q.
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So you get 1 minus q here.
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And now this term becomes 6q
times 1 minus q squared.
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And so really, the bottom is
also 6q times 1 minus q
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squared dq.
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And if you go through the same
sort of calculus, what you get
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is that, in this case,
the answer is
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12q 1 minus q squared.
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So let me rewrite what
the first case was.
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The first case, when x equals
1 was equal to 12q
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squared 1 minus q.
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So they look quite similar,
except that this one has q
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squared, this one has
1 minus q squared.
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And if you take this one, the
case where you observe a 0,
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and you plot that, it
turns out it looks
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something like this.
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And this actually doesn't look
like it, but it should be the
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peak is at 1/3.
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And so you notice, first of all,
that these are symmetric.
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There's some symmetry
going on.
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And this interpretation in this
case is that, because you
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observed 0, which corresponds to
observing tails, that gives
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you evidence that the bias of
the coin is relatively low, or
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the probability of getting
heads with this coin is
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relatively low, which pushes
your belief about q towards
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the smaller values.
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OK, so it turns out that
this distribution, this
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distribution, and the original
distribution of q, all fall
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under family of distributions
called the beta distribution.
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And they're parameterized by
a couple of parameters.
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And it's used frequently to
model things like the bias of
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a coin, or anything that's a
random variable that's bounded
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between 0 and 1.
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OK, so this problem allowed us
to get some more exercise with
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Bayes' rule, this continuous
discrete
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version of Bayes' rule.
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And if you go through all the
steps, you'll find that it's
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relatively straightforward to
go through the formula and
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plug in the different parts of
the terms and go through a
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little bit of calculus
and find the answer.
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And it's always good to go
back, once you find the
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answer, to look at it a little
bit and make sure that
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actually makes sense.
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So you can convince yourself
that it's not
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something that looks--
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